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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.20190 |
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| _version_ | 1866910111674400768 |
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| author | Hirotsu, Takashi |
| author_facet | Hirotsu, Takashi |
| contents | Let $n,$ $m \geq 2$ be integers, and let $R$ be a subring of $\mathbb R$ with field of fractions $F.$ In this article, we generalize the rational angle bisection problem previously proposed by the author to the following problem: which linearly independent vectors $\boldsymbol{a},$ $\boldsymbol{b} \in R^n$ form an angle with a sequence of $m$-sector vectors lying in $R^n$? When $\boldsymbol{a}$ and $\boldsymbol{b}$ are nonorthogonal, we prove that this condition is equivalent to the existence of a root in $F$ of a certain $m$-th degree polynomial over $R.$ In particular, when $R = \mathbb Z,$ the condition holds if and only if the polynomial has a root among the divisors of its constant term. When $m = 2^e$ with an integer $e \geq 1,$ we also prove that the condition is equivalent to $\cos (θ/2^{e-1}) \in F,$ where $θ$ is the angle between $\boldsymbol{a}$ and $\boldsymbol{b}.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20190 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Algebraic Characterizations of Angle Multisections over Rings Hirotsu, Takashi Number Theory Metric Geometry 11C08 (Primary) 51N20, 51M16 (Secondary) Let $n,$ $m \geq 2$ be integers, and let $R$ be a subring of $\mathbb R$ with field of fractions $F.$ In this article, we generalize the rational angle bisection problem previously proposed by the author to the following problem: which linearly independent vectors $\boldsymbol{a},$ $\boldsymbol{b} \in R^n$ form an angle with a sequence of $m$-sector vectors lying in $R^n$? When $\boldsymbol{a}$ and $\boldsymbol{b}$ are nonorthogonal, we prove that this condition is equivalent to the existence of a root in $F$ of a certain $m$-th degree polynomial over $R.$ In particular, when $R = \mathbb Z,$ the condition holds if and only if the polynomial has a root among the divisors of its constant term. When $m = 2^e$ with an integer $e \geq 1,$ we also prove that the condition is equivalent to $\cos (θ/2^{e-1}) \in F,$ where $θ$ is the angle between $\boldsymbol{a}$ and $\boldsymbol{b}.$ |
| title | Algebraic Characterizations of Angle Multisections over Rings |
| topic | Number Theory Metric Geometry 11C08 (Primary) 51N20, 51M16 (Secondary) |
| url | https://arxiv.org/abs/2602.20190 |